Time-dependent perturbation theory for vibrational energy relaxation and dephasing in peptides and proteins

Without invoking the Markov approximation, we derive formulas for vibrational energy relaxation (VER) and dephasing for an anharmonic system oscillator using a time-dependent perturbation theory. The system-bath Hamiltonian contains more than the third order coupling terms since we take a normal mode picture as a zeroth order approximation. When we invoke the Markov approximation, our theory reduces to the Maradudin-Fein formula which is used to describe the VER properties of glass and proteins. When the system anharmonicity and the renormalization effect due to the environment vanishes, our formulas reduce to those derived by and Mikami and Okazaki [J. Chem. Phys. 121, 10052 (2004)] invoking the path-integral influence functional method with the second order cumulant expansion. We apply our formulas to VER of the amide I mode of a small amino-acid like molecule, N-methylacetamide, in heavy water.     

Simulation of vibrational energy transfer in two-dimensional infrared spectroscopy of amide I and amide II modes in solution

Population transfer between vibrational eigenstates is important for many phenomena in chemistry. In solution, this transfer is induced by fluctuations in molecular conformation as well as in the surrounding solvent. We develop a joint electrostatic density functional theory map that allows us to connect the mixing of and thereby the relaxation between the amide I and amide II modes of the peptide building block N-methyl acetamide. This map enables us to extract a fluctuating vibrational Hamiltonian from molecular dynamics trajectories. The linear absorption spectrum, population transfer, and two-dimensional infrared spectra are then obtained from this Hamiltonian by numerical integration of the Schrodinger equation. We show that the amide I/amide II cross peaks in two-dimensional infrared spectra in principle allow one to follow the vibrational population transfer between these two modes. Our simulations of N-methyl acetamide in heavy water predict an efficient relaxation between the two modes with a time scale of 790 fs. This accounts for most of the relaxation of the amide I band in peptides, which has been observed to take place on a time scale of 450 fs in N-methyl acetamide. We therefore conclude that in polypeptides, energy transfer to the amide II mode offers the main relaxation channel for the amide I vibration.     

Vibrational energy relaxation of a diatomic molecule in a room-temperature ionic liquid

Vibrational energy relaxation (VER) dynamics of a diatomic solute in ionic liquid 1-ethyl-3-methylimidazolium hexafluorophosphate (EMI(+)PF(6) (-)) are studied via equilibrium and nonequilibrium molecular dynamics simulations. The time scale for VER is found to decrease markedly with the increasing solute dipole moment, consonant with many previous studies in polar solvents. A detailed analysis of nonequilibrium results shows that for a dipolar solute, dissipation of an excess solute vibrational energy occurs almost exclusively via the Lennard-Jones interactions between the solute and solvent, while an oscillatory energy exchange between the two is mainly controlled by their electrostatic interactions. Regardless of the anharmonicity of the solute vibrational potential, VER becomes accelerated as the initial vibrational energy increases. This is attributed primarily to the enhancement in variations of the solvent force on the solute bond, induced by large-amplitude solute vibrations. One interesting finding is that if a time variable scaled with the initial excitation energy is employed, dissipation dynamics of the excess vibrational energy of the dipolar solute tend to show a universal behavior irrespective of its initial vibrational state. Comparison with water and acetonitrile shows that overall characteristics of VER in EMI(+)PF(6) (-) are similar to those in acetonitrile, while relaxation in water is much faster than the two. It is also found that the Landau-Teller theory predictions for VER time scale obtained via equilibrium simulations of the solvent force autocorrelation function are in reasonable agreement with the nonequilibrium results.     

Vibrational relaxation pathways of amide I and amide II modes in N-methylacetamide

We studied the vibrational energy relaxation mechanisms of the amide I and amide II modes of N-methylacetamide (NMA) monomers dissolved in bromoform using polarization-resolved femtosecond two-color vibrational spectroscopy. The results show that the excited amide I vibration transfers its excitation energy to the amide II vibration with a time constant of 8.3 ± 1 ps. In addition to this energy exchange process, we observe that the excited amide I and amide II vibrations both relax to a final thermal state. For the amide I mode this latter process dominates the vibrational relaxation of this mode. We find that the vibrational relaxation of the amide I mode depends on frequency which can be well explained from the presence of two subbands with different vibrational lifetimes (~1.1 ps on the low frequency side and ~2.7 ps on the high frequency side) in the amide I absorption spectrum.     

Comparison between the Landau-Teller and flux-flux methods for computing vibrational energy relaxation rate constants in the condensed phase

The calculation of vibrational energy relaxation (VER) rate constants in the condensed phase is usually based on the Landau-Teller formula, which puts them in terms of the Fourier transform, at the vibrational frequency, of the autocorrelation function of the force exerted on the relaxing mode by the bath modes. An alternative expression for the VER rate constant puts it in terms of the autocorrelation function of the vibrational energy flux. In this paper, we compare the predictions obtained via those two methods in the case of iodine in liquid xenon. We find that the computational cost underlying both methods is comparable and that they predict similar VER rates. However, while the calculation of the VER rate via the Landau-Teller formula is somewhat more direct, the predictions obtained via the flux-flux formula are in somewhat better agreement with the VER rates obtained from nonequilibrium molecular dynamics simulations.     

Nonperturbative vibrational energy relaxation effects on vibrational line shapes

A general formulation of nonperturbative quantum dynamics of solutes in a condensed phase is proposed to calculate linear and nonlinear vibrational line shapes. In the weak solute-solvent interaction limit, the temporal absorption profile can be approximately factorized into the population relaxation profile from the off-diagonal coupling and the pure-dephasing profile from the diagonal coupling. The strength of dissipation and the anharmonicity-induced dephasing rate are derived in Appendix A. The vibrational energy relaxation (VER) rate is negligible for slow solvent fluctuations, yet it does not justify the Markovian treatment of off-diagonal contributions to vibrational line shapes. Non-Markovian VER effects are manifested as asymmetric envelops in the temporal absorption profile, or equivalently as side bands in the frequency domain absorption spectrum. The side bands are solvent-induced multiple-photon effects which are absent in the Markovian VER treatment. Exact path integral calculations yield non-Lorentzian central peaks in absorption spectrum resulting from couplings between population relaxations of different vibrational states. These predictions cannot be reproduced by the perturbative or the Markovian approximations. For anharmonic potentials, the absorption spectrum shows asymmetric central peaks and the asymmetry increases with anharmonicity. At large anharmonicities, all the approximation schemes break down and a full nonperturbative path integral calculation that explicitly accounts for the exact VER effects is needed. A numerical analysis of the O-H stretch of HOD in D(2)O solvent reveals that the non-Markovian VER effects generate a small recurrence of the echo peak shift around 200 fs, which cannot be reproduced with a Markovian VER rate. In general, the nonperturbative and non-Markovian VER contributions have a stronger effect on nonlinear vibrational line shapes than on linear absorption.     

Classical vs quantum vibrational energy relaxation pathways in solvated polyatomic molecules

Vibrational energy relaxation (VER) of solvated polyatomic molecules can occur via different pathways. In this paper, we address the question of whether treating VER classically or quantum-mechanically can lead to different predictions with regard to the preferred pathway. To this end, we consider the relaxation of the singly excited asymmetric stretch of a rigid, symmetrical, and linear triatomic molecule (A-B-A) in a monatomic liquid. In this case, VER can occur either directly to the ground state or indirectly via intramolecular vibrational relaxation (IVR) to the symmetric stretch. We have calculated the rates of these two different VER pathways via classical mechanics and the linearized semiclassical (LSC) method. When the mass of the terminal A atoms is significantly larger than that of the central B atom, we find that LSC points to intermolecular VER as the preferred pathway, whereas the classical treatment points to IVR. The origin of this trend reversal appears to be purely quantum-mechanical and can be traced back to the significantly weaker quantum enhancement of solvent-assisted IVR in comparison to that of intermolecular VER.     

Vibrational energy relaxation of polyatomic molecules in liquid solution via the linearized semiclassical method

Vibrational energy relaxation (VER) of polyatomic, as opposed to diatomic, molecules can occur via different, often solvent assisted, intramolecular and/or intermolecular pathways. In this paper, we apply the linearized semiclassical (LSC) method for calculating VER rates in the prototypical case of a rigid, symmetrical and linear triatomic molecule (A-B-A) in a monatomic liquid. Starting at the first excited state of either the symmetric or asymmetric stretches, VER can occur either directly to the ground state or indirectly via intramolecular vibrational relaxation (IVR). The VER rate constants for the various pathways are calculated within the framework of the Landau-Teller formalism, where they are expressed in terms of two-time quantum-mechanical correlation functions. The latter are calculated by the LHA-LSC method, which puts them in a "Wignerized" form, and employs a local harmonic approximation (LHA) in order to compute the necessary multidimensional Wigner integrals. Results are reported for the LHL/Ar model of Deng and Stratt [J. Chem. Phys. 2002, 117, 1735], as well as for CO(2) in liquid argon and in liquid neon. The LHA-LSC method is shown to give rise to significantly faster VER and IVR rates in comparison to the classical treatment, particularly at lower temperatures. We also find that the type and extent of the quantum rate enhancement is strongly dependent on the particular VER pathway. Finally, we find that the classical and semiclassical treatments can give rise to opposite trends when it comes to the dependence of the VER rates on the solvent.     

Fluctuating energy level Landau-Teller theory: application to the vibrational energy relaxation of liquid methanol

State-to-state vibrational energy relaxation (VER) rates of the OH-stretch fundamental to select vibrational modes of liquid methanol are presented. The rates are calculated via a modified, fluctuating Landau-Teller (FLT) theory approach, which allow for dynamical vibrational energy level shifts. These rates are then compared to previously published results from Gulmen and Sibert [J. Phys. Chem. A 2004, 108, 2389] for the traditional Landau-Teller (LT) method as well as results calculated through time-dependent perturbation theory (TD), which naturally allow for the fluctuation. For the first time, this method is applied to a polyatomic molecular system, and the FLT theory greatly reduces the discrepancy between the LT and TD results or, at a minimum, is comparable to the LT approach with very little additional computational cost.     

The anharmonic vibrational potential and relaxation pathways of the amide I and II modes of N-methylacetamide

We investigate the influence of isotopic substitution and solvation of N-methylacetamide (NMA) on anharmonic vibrational coupling and vibrational relaxation of the amide I and amide II modes. Differences in the anharmonic potential of isotopic derivatives of NMA in D2O and DMSO-d6 are quantified by extraction of the anharmonic parameters and the transition dipole moment angles from cross-peaks in the two-dimensional infrared (2D-IR) spectra. To interpret the effects of isotopic substitution and solvent interaction on the anharmonic potential, density functional theory and potential energy distribution calculations are performed. It is shown that the origin of anharmonic variation arises from differing local mode contributions to the normal modes of the NMA isotopologues, particularly in amide II. The time domain manifestation of the coupling is the coherent exchange of excitation between amide modes seen as the quantum beats in femtosecond pump-probes. The biphasic behavior of population relaxation of the pump-probe and 2D-IR experiments can be understood by the rapid exchange of strongly coupled modes within the peptide backbone, followed by picosecond dissipation into weakly coupled modes of the bath.     

Vibrational relaxation pathways of AI and AII modes in N-methylacetamide clusters

We studied the pathways of vibrational energy relaxation of the amide I (~1660 cm?1) and amide II (~1560 cm?1) vibrational modes of N-methylacetamide (NMA) in CCl? solution using two-color femtosecond vibrational spectroscopy. We measured the transient spectral dynamics upon excitation of each of these amide modes. The results show that there is no energy transfer between the amide I (AI) and amide II (AII) modes. Instead we find that the vibrational energy is transferred on a picosecond time scale to a common combination tone of lower-frequency modes. By use of polarization-resolved femtosecond pump-probe measurements we also study the reorientation dynamics of the NMA molecules and the relative angle between the transition dipole moments of the AI and AII vibrations. The spectral dynamics at later times after the excitation (>40 ps) reveal the presence of a dissociation process of the NMA aggregates, trimers, and higher order structures into dimers and monomers. By measuring the dissociation kinetics at different temperatures, we determined the activation energy of this dissociation E(a) = 35 ± 3 kJ mol?1.     

Energy relaxation versus spectral diffusion of the OH-stretching vibration of HOD in liquid-to-supercritical deuterated water

The dynamics of vibrational energy relaxation (VER) of the OH-stretching vibration of HOD in liquid-to-supercritical heavy water is studied as a function of temperature and solvent density by femtosecond mid-infrared spectroscopy. Using the dielectric constant of the fluid both, the OH-stretching absorption frequency and the VER rate, can be correlated phenomenologically with the average hydrogen-bond connectivity within the random D2O network. This correlation enables the identification of thermodynamic conditions under which spectral diffusion due to hydrogen-bond breakage/formation is much faster than VER.     

Vibrational relaxation of azide ions in liquid-to-supercritical water

The dynamics of vibrational energy relaxation (VER) of the aqueous azide anion was studied over a wide temperature (300 K ≤ T ≤ 663 K) and density (0.6 g cm(-3) ≤ ρ ≤ 1.0 g cm(-3)) range thereby covering the liquid and the supercritical phase of the water solvent. Femtosecond mid-infrared spectroscopy on the ν(3) band associated with the asymmetric stretching vibration of the azide anion was used to monitor the relaxation dynamics in a time-resolved fashion. The variation of the vibrational relaxation rate constant with temperature and density was found to be rather small. Surprisingly, the simple isolated binary collision model is able to fully reproduce the experimentally observed temperature and density dependence of the relaxation rate provided a local density correction around the vibrationally excited solute based on classical molecular dynamics simulations is used. The simulations further suggest that head-on collisions of the solvent with the terminal nitrogen atoms rather than side-on collisions with the central nitrogen atom of the azide govern the vibrational energy relaxation of this system. Finally, the importance of hydrogen bonding for the VER dynamics in this system is briefly discussed.     

Direct dynamics study of ultrafast vibrational energy relaxation in ice Ih

Car-Parrinello molecular dynamics (CPMD) and a previously developed wave packet model are used to study ultrafast relaxation in water clusters. Water clusters of 15 water molecules are used to represent ice Ih. The relaxation is studied by exciting a symmetric or an asymmetric stretch mode of the central water molecule. The CPMD results suggest that relaxation occurs within 100 fs. This is in agreement with experimental work by Woutersen and Bakker and the earlier wave packet calculations. The CPMD results further indicate that the excitation energy is transferred both intramolecularly and intermolecularly on roughly the same time scale. The intramolecular energy transfer occurs predominantly between the symmetric and asymmetric modes while the bend mode is largely left unexcited on the short time scale studied here.     

Vibrational energy relaxation of benzene dimer and trimer in the CH stretching region studied by picosecond time-resolved IR-UV pump-probe spectroscopy

Vibrational energy relaxation (VER) of the Fermi polyads in the CH stretching vibration of the benzene dimer (Bz(2)) and trimer (Bz(3)) has been investigated by picosecond (ps) time-resolved IR-UV pump-probe spectroscopy in a supersonic beam. The vibrational bands in the 3000-3100 cm(-1) region were excited by a ps IR pulse and the time evolutions at the pumped and redistributed (bath) levels were probed by resonance enhanced multiphoton ionization with a ps UV pulse. For Bz(2), a site-selective excitation in the T-shaped structure was achieved by using the isotope-substituted heterodimer hd, where h = C(6)H(6) and d = C(6)D(6), and its result was compared with that of hh homodimer. In the hd heterodimer, the two isomers, h(stem)d(top) and h(top)d(stem), show remarkable site-dependence of the lifetime of intracluster vibrational energy redistribution (IVR); the lifetime of the Stem site [h(stem)d(top), 140-170 ps] is ~2.5 times shorter than that of the Top site [h(top)d(stem), 370-400 ps]. In the transient UV spectra, a broad electronic transition due to the bath modes emerges and gradually decays with a nanosecond time scale. The broad transition shows different time profile depending on UV frequency monitored. These time profiles are described by a three-step VER model involving IVR and vibrational predissociation: initial → bath1(intramolecular) → bath2(intermolecular) → fragments. This model also describes well the observed time profile of the Bz fragment. The hh homodimer shows the stepwise VER process with time constants similar to those of the hd dimer, suggesting that the excitation-exchange coupling of the vibrations between the two sites is very weak. Bz(3) also exhibited the stepwise VER process, though each step is faster than Bz(2).     

Vibrational and rotational dynamics of cyanoferrates in solution

Ultrafast infrared spectroscopy has been used to measure vibrational energy relaxation (VER) and reorientation (Tr) times for the high frequency vibrational bands of potassium ferrocyanide and ferricyanide (CN stretches), and sodium nitroprusside (SNP, CN, and NO stretches) in water and several other solvents. Relatively short VER times (4-43 ps) are determined for the hexacyano species and for the NO band of SNP, but the CN band of SNP relaxes much more slowly (55-365 ps). The solvent dependence of the VER times is similar for all the solutes and resembles what has been previously observed for triatomic molecular ions [Li et al., J. Chem. Phys. 98, 5499 (1993)]. Anisotropy decay times are also measured from the polarization dependence of the transient absorptions. The Tr times determined for SNP are different for the different vibrational bands; for the nondegenerate NO mode of nitroprusside (SNP) they are much longer (>15 ps), correlate with solvent viscosity, and are attributed to overall molecular rotation. The short Tr (<10 ps) times for the CN band in SNP and for the hexacyanoferrates are due to dipole orientational relaxation in which the transition moment rapidly redistributes among the degenerate modes. There is no evidence of intramolecular vibrational relaxation (IVR) to other high frequency modes. VER times measured for hexacarbonyls and SNP in methanol are similar, which suggests that the generally faster VER for the latter is in part because they are soluble in more strongly interacting polar solvents. The results are compared to those for small ions and metal carbonyls and are discussed in terms of the importance of solute charge and symmetry on VER.     

Vibrational relaxation in simulated two-dimensional infrared spectra of two amide modes in solution

Two-dimensional infrared spectroscopy is capable of following the transfer of vibrational energy between modes in real time. We develop a method to include vibrational relaxation in simulations of two-dimensional infrared spectra at finite temperature. The method takes into account the correlated fluctuations that occur in the frequencies of the vibrational states and in the coupling between them as a result of interaction with the environment. The fluctuations influence the two-dimensional infrared line shape and cause vibrational relaxation during the waiting time, which is included using second-order perturbation theory. The method is demonstrated by applying it to the amide-I and amide-II modes in N-methylacetamide in heavy water. Stochastic information on the fluctuations is obtained from a molecular dynamics trajectory, which is converted to time dependent frequencies and couplings with a map from a density functional calculation. Solvent dynamics with the same frequency as the energy gap between the two amide modes lead to efficient relaxation between amide-I and amide-II on a 560 fs time scale. We show that the cross peak intensity in the two-dimensional infrared spectrum provides a good measure for the vibrational relaxation.     

Vibrational energy relaxation rates of H2 and D2 in liquid argon via the linearized semiclassical method

The vibrational energy relaxation (VER) rates for H2 and D2 in liquid argon (T=152 K, rho=1.45x1022 cm-3) are calculated using the linearized semiclassical (LSC) method (J. Phys. Chem. 2003, 107, 9059, 9070). The calculation is based on Fermi's golden rule. The VER rate constant is expressed in terms of the quantum-mechanical force-force correlation function, which is then estimated using the LSC method. A local harmonic approximation (LHA) is employed in order to compute the multidimensional Wigner integrals underlying the LSC approximation. The H2-Ar and D2-Ar interactions are described by the three-body potential of Bissonette et al. (J. Phys. Chem. A 1996, 105, 2639). The LHA-LSC-based VER rate constants for both D2 and H2 are found to be about 2-3 orders of magnitude slower than those obtained experimentally. However, their ratio agrees quantitatively with the corresponding experimental result. In contrast, the classical VER rate constants are found to be 8-9 orders of magnitude slower than those obtained experimentally, and their ratio is found to be qualitatively different from the corresponding experimental result.     

Vibrational relaxation of azide in formamide reverse micelles

Static and ultrafast infrared spectroscopy have been used to measure absorption spectra and vibrational energy relaxation (VER) times for the antisymmetric stretching vibrational band of azide, N(3)(-), in formamide-containing reverse micelles (RMs). RMs were formed in n-heptane using the surfactant AOT, sodium bis(2-ethylhexyl) sulfosuccinate. The VER times were found to be significantly longer than in bulk formamide. The VER times became longer as the molar ratio of formamide to AOT, omega(F), was decreased. Decreasing omega(F) also resulted in substantial blue shifts of the azide static absorption band compared to the frequency in bulk formamide. The omega(F) dependent studies are consistent with expected size trends, where a larger RM results in more bulklike polar solvent and faster VER rates. These results are in contrast to aqueous AOT RMs where VER times were indistinguishable from those in the bulk and the static spectral shifts were much smaller. The differences between the static and dynamic behavior in aqueous and formamide RMs are related to differences in structural changes upon confinement in RMs.     

Vibrational energy relaxation of the bend fundamental of dilute water in liquid chloroform and d-chloroform

The population lifetimes of the bend fundamental of dilute water in liquid chloroform (8.5 ps) and d-chloroform (28.5 ps) display an interesting solvent isotope effect. As the lowest excited vibrational state of the molecule, the water bend fundamental relaxes directly to the ground state with about 1600 cm-1 of energy released to the other degrees of freedom. The strong solvent isotope effect along with the large energy gap indicates the participation of solvent vibrational modes in this vibrational energy relaxation process. We calculate the vibrational energy relaxation rates of the water bend in chloroform and d-chloroform using the Landau-Teller formula with a new potential model developed and parametrized self-consistently to describe the chloroform-water interaction. The computed values are in reasonable agreement with the experimental results, and the trend for the isotope effect is correct. It is found that energy transfer to the solvent vibrations does indeed play an important role. Nevertheless, no single dominant solvent accepting mode can be identified; the relaxation appears to involve both the bend and the C-Cl stretches, and frequency changes of all of these modes upon deuteration contribute to the observed solvent isotope effect.